Showing papers on "Free product published in 2008"
TL;DR: The first L 2 -Betti number for certain amalgamated free products of groups is given in this article, where the expected free entropy dimension of Popa algebra generators of free group factors is shown to agree for generating sets of many groups.
Abstract: We calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von Neumann algebras, with amalgamation over a hyperfinite subalgebra. In particular, some ‘exotic’ Popa algebra generators of free group factors are shown to have the expected free entropy dimension. We also show that microstates and non-microstates free entropy dimension agree for generating sets of many groups. In the appendix, the first L 2 -Betti number for certain amalgamated free products of groups is calculated.
74 citations
TL;DR: In this article, the unit ball of the Gromov-thurston norm on H2(G; ℝ) is a finite-sided rational polyhedron.
Abstract: Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H2(G; ℚ) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on H2(G; ℝ) is a finite-sided rational polyhedron.
43 citations
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TL;DR: In this article, the maximal subgroups of the free idempotent generated semigroup on a biordered set were studied and shown to be isomorphic to the free abelian group of rank 2.
Abstract: We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup.
34 citations
TL;DR: In this paper, the model-theoretic properties of the nonabelian free group were studied in the light of Sela's recent result [15] on stability and results announced by Bestvina and Feighn on "negligible subsets" of free groups.
Abstract: We study model-theoretic and stability-theoretic properties of the nonabelian free group in the light of Sela’s recent result [15] on stability and results announced by Bestvina and Feighn on “negligible subsets” of free groups. We point out analogies between the free group and socalled bad groups of finite Morley rank, and prove “non CM-triviality” of the free group.
30 citations
01 May 2008
TL;DR: In this article, a family of groups and a free group are considered, and the free product of the family is the product of all the groups in the family and the group.
Abstract: Let (Gi j i 2 I) be a family of groups, let F be a free group, and let G = F ⁄ ⁄ i2I Gi, the free product of F and all the Gi
29 citations
TL;DR: Several transfer results for rational subsets and finitely generated subgroups of HNN-extensions G = 〈 H,t; t-1 at = φ(a) (a ∈ A) 〉 and amalgamated free products G = H *A J such that the associated subgroup A is finite.
Abstract: Several transfer results for rational subsets and finitely generated subgroups of HNN-extensions G = 〈 H,t; t-1 at = φ(a) (a ∈ A) 〉 and amalgamated free products G = H *A J such that the associated subgroup A is finite. These transfer results allow to transfer decidability properties or structural properties from the subgroup H (resp. the subgroups H and J) to the group G.
27 citations
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TL;DR: In this paper, a family of automata with n states, n>3, acting on a rooted binary tree that generate the free products of cyclic groups of order 2 was constructed.
Abstract: We construct a family of automata with n states, n>3, acting on a rooted binary tree that generate the free products of cyclic groups of order 2.
27 citations
TL;DR: In this paper, it was shown that if the pair (Γ, Γ_0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ_ 0) is a singular MASA in M = L(L) which satisfies a weakly mixing condition.
Abstract: Let Γ be a countable group and let Γ_0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ_0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ_0) is a singular MASA in M=L(Γ) which satisfies a weakly mixing condition. If, moreover, it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, Higman–Neumann–Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.
18 citations
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11 Mar 2008
TL;DR: In this paper, it was shown that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the finite-by-cyclic subgroup.
Abstract: We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.
17 citations
TL;DR: For any positive integer n, A n is the class of all groups G such that, for 0 √ i ⩽ n, H i (G ˆ, A ) ≅ H i(G, A ) for every finite discrete G ˀ -module A.
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TL;DR: In this article, a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial-isometries need not have trivial intersection was introduced.
Abstract: We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the $C^*$-algebras and the non-selfadjoint operator algebras associated to edge-colored directed graphs. These algebras arise as free products of directed graph algebras with amalgamation. We then determine the $C^*$-envelopes for a large class of the non-selfadjoint algebras. Finally, we relate properties of the edge-colored directed graphs to properties of the associated $C^*$-algebra, including simplicity and nuclearity. Using the free product description of these algebras we investigate the $K$-theory of these algebras.
TL;DR: In this paper, the condition of a profinite group being semi-free was introduced, which is more general than being free and more restrictive than being quasi-free, and the usual permanence properties of free groups carry over to semi free groups, and it was shown that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.
Abstract: We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.
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TL;DR: In this article, it was shown that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa.
Abstract: We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in particular the first examples of such actions for the free group on $2$ generators.
TL;DR: In this article, the authors generalize the results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively-hierarchical groups, and prove the following: if G is a nonelementary relatively-hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group.
Abstract: We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).
TL;DR: In this article, Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups.
Abstract: Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture
TL;DR: In this paper, a cross product algebra with N-generators is considered and the moments and cumulants of operators in the crossed product are computed for groups of all automorphisms.
Abstract: In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space Hand G, a group and let : G AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product G of M and G with respect to is a von Neumann algebra acting on , generated by M and , where is the unitary representation of g on . We show that . We compute moments and cumulants of operators in . By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if is the free group with N-generators, then the crossed product algebra satisfies that , whenerver .
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TL;DR: In this article, it was shown that if G is the free product of Z and infinitely many copies of a non-trivial group \Gamma, then S_factor(G)=S_eqrel(G)={1}.
Abstract: Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group \Gamma, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=\Gamma*\Lambda, with \Gamma, \Lambda finitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.
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TL;DR: In this paper, a new concept of topological orbit dimension of $n$-tuples of elements in a unital C$^*$ algebra was introduced, and it was shown that the Voiculescu's topological free entropy dimension of any family of self-adjoint generators of a nuclear C$ *$ algebra is less than or equal to 1.
Abstract: In the paper, we introduce a new concept of topological orbit dimension of $n$-tuples of elements in a unital C$^*$ algebra. Using this concept, we conclude that the Voiculescu's topological free entropy dimension of any family of self-adjoint generators of a nuclear C$^*$ algebra is less than or equal to 1. We also show that the topological free entropy dimension is additive in the full free products of unital C$^*$ algebras. In the appendix, we show that unital full free product of Blackadar and Kirchberg's unital MF algebras is also MF algebra.
TL;DR: In this article, the minimum number of relators required to generate a finite cyclic group with positive relation gap is shown to be 4, and it seems plausible that the ratio of the relators of G should be 4.
Abstract: Abstract Let G be a group of the form , the free product of n subgroups, and let M be a ℤG-module of the form . We shall give formulae in various situations for dℤG (M), the minimum number of elements required to generate M. In particular if C 1, C 2 are non-trivial finite cyclic groups of coprime orders, and F/R ≅ G is the free presentation obtained from the natural free presentations of the two factors, then the number of generators of the relation module, dℤG (R/R′), is 3. It seems plausible that the minimum number of relators of G should be 4, and this would give a finitely presented group with positive relation gap. However we cannot prove this last statement.
TL;DR: In this paper, it was shown that the fundamental group of a closed topological n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation.
Abstract: A closed topological n-manifold Mn is of S1-category 2 if it can be covered by two open subsets W1,W2 such that the inclusions Wi → Mn factor homotopically through maps Wi → S1 → Mn. We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.
04 Sep 2008
TL;DR: In this article, the commensurability classifications of free products of finitely many finitely generated abelian groups are given and shown to coincide with the quasi-isometry classification.
Abstract: We give the commensurability classifications of free products of finitely many finitely generated abelian groups. We show this coincides with the quasi-isometry classification and prove that this class of groups is quasi-isometrically rigid.
TL;DR: In this article, it was shown that if Gj are right orderable groups with non-trivial cyclic subgroups Cj (j = 1, 2), then the (group) free product of G1 and G2 with C1 and C2 amalgamated is right-orderable.
Abstract: We prove that if Gj are right orderable groups with non-trivial cyclic subgroups Cj (j = 1, 2), then the (group) free product of G1 and G2 with C1 and C2 amalgamated is right orderable. We will deduce this from our main theorem that every lattice-ordered group can be embedded in one in which there are precisely four conjugacy classes. As a consequence of our method, we also show that every right ordered group can be embedded in a right ordered group in which any two non-identity elements are conjugate. —————————————– AMS Classification: 06F15, 20F60, 20B27, 20F10.
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TL;DR: In this article, it was shown that the compressed word problem for an HNN-extension with base group H and finite associated subgroups is polynomial time Turing-reducible.
Abstract: It is shown that the compressed word problem for an HNN-extension with base group H and finite associated subgroups is polynomial time Turing-reducible to the compressed word problem for H. An analogous result for amalgamated free products is shown as well.
TL;DR: In this article, the authors studied percolation on the Cayley graph of a free product of groups and showed that the critical probability of the free product G1 * G2 * ⋯ * Gn of groups is 0.5199, the unique root of the polynomial 2p5 - 6p4 + 2p3 + 4p2 -1 in the interval (0, 1).
Abstract: In this article we study percolation on the Cayley graph of a free product of groups. The critical probability pc of a free product G1 * G2 * ⋯ * Gn of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G1, G2, …, Gn. For finite groups this equation is polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that pc for the Cayley graph of the modular group PSL2(ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p5 - 6p4 + 2p3 + 4p2 - 1 in the interval (0, 1). In the case when groups Gi can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of G1 * G2 * ⋯ * Gn and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, pexp of the free product is just the minimum of pexp for the factors.
TL;DR: In this paper, Singer et al. showed that in a free product of separable unital C*-algebras, states on each algebra can be simultaneously extended to a pure state on the free product.
Abstract: We answer two questions raised by I. M. Singer concerning free products. We prove that in a free product of separable unital C*-algebras, states on each algebra can be simultaneously extended to a pure state on the free product. We also show that the second dual of the free product of unital C*-algebras is the von Neumann algebra free product of their second duals. We give a proof that the extreme points of the set of tracial states of a C*-algebra is the set of factor tracial states. Mathematics subject classification (2000): 46L09, 46L30, 46L05.
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TL;DR: In this paper, it was shown that the reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are not quasidiagonal.
Abstract: In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup and Thorbjorsen's work on the reduced C ∗ -algebras of free
01 May 2008
TL;DR: In this paper, it was shown that for an action of a finite group G on a systolic complex X there exists a G-invariant subcomplex of X of diameter = 5.
Abstract: We prove that for an action of a finite group G on a systolic complex X there exists a G-invariant subcomplex of X of diameter =5. For 7-systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7-systolic groups over finite subgroups are also 7-systolic.
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TL;DR: In this article, it was shown that the amalgamated free product of two free groups of rank two over a common cyclic subgroup admits an amenable, faithful, transitive action on an infinite countable set.
Abstract: We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup, admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such an action, which applies for example to surface groups and fundamental groups of surface bundles over $\mathbb{S}^1$.
TL;DR: In this article, the structure of the cohomology ring of an almost-direct product of free groups is determined and used to analyze the topological complexity of the associated Eilenberg-Mac Lane space.
Abstract: An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space.